Erdős and Sós conjectured in 1962 that if the average degree of a graph G exceeds k − 2, then G contains every tree on k vertices. In 1984, Zho proved the special case where G has k vertices. In 1996, Woźniak proved the cases where G has k + 2 vertices. We prove the conjecture for the case where G has k + 3 vertices.

We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and deletes vertices randomly. At time t, with probability α1 > 0 we add a new vertex ut and m random edges incident with ut. The neighbours of ut are chosen with probability proportional to degree. With probability α − α1 ≥ 0 we add m random edges to existing vertices where the endpoints ar...

For a graph G, let a(G) denote the maximum size of a subset of vertices that induces a forest. Suppose that G is connected with n vertices, e edges, and maximum degree ∆. Our results include: (a) if ∆ ≤ 3, and G 6= K4, then a(G) ≥ n−e/4−1/4 and this is sharp for all permissible e ≡ 3 (mod 4), (b) if ∆ ≥ 3, then a(G) ≥ α(G) + (n − α(G))/(∆ − 1)2. Several problems remain open.

Let G be a graph of order n, and let n = P k i=1 a i be a partition of n with a i 2. Let v 1 ; : : : ; v k be given distinct vertices of G. Suppose that the minimum degree of G is at least 3k. In this paper, we prove that there exists a decomposition of the vertex set V (G) = S k i=1 A i such that jA i j = a i , v i 2 A i , and the subgraph induced by A i contains no isolated vertices for all i...

For a graph G, the irregularity and total irregularity of G are defined as irr(G)=∑_(uv∈E(G))〖|d_G (u)-d_G (v)|〗 and irr_t (G)=1/2 ∑_(u,v∈V(G))〖|d_G (u)-d_G (v)|〗, respectively, where d_G (u) is the degree of vertex u. In this paper, we characterize all ‎connected Eulerian graphs with the second minimum irregularity, the second and third minimum total irregularity value, respectively.

The Zagreb indices are the oldest graph invariants used in mathematical chemistry to predict the chemical phenomena. In this paper we define the multiple versions of Zagreb indices based on degrees of vertices in a given graph and then we compute the first and second extremal graphs for them.

‎the gutman index and degree distance of a connected graph $g$ are defined as‎ ‎begin{eqnarray*}‎ ‎textrm{gut}(g)=sum_{{u,v}subseteq v(g)}d(u)d(v)d_g(u,v)‎, ‎end{eqnarray*}‎ ‎and‎ ‎begin{eqnarray*}‎ ‎dd(g)=sum_{{u,v}subseteq v(g)}(d(u)+d(v))d_g(u,v)‎, ‎end{eqnarray*}‎ ‎respectively‎, ‎where‎ ‎$d(u)$ is the degree of vertex $u$ and $d_g(u,v)$ is the distance between vertices $u$ and $v$‎. ‎in th...

the total irregularity of a graph g is defined as 〖irr〗_t (g)=1/2 ∑_(u,v∈v(g))▒〖|d_u-d_v |〗, where d_u denotes the degree of a vertex u∈v(g). in this paper by using the gini index, we obtain the ordering of the total irregularity index for some classes of connected graphs, with the same number of vertices.

This paper presents an algorithm for finding edge-disjoint paths in a given plane grid bounded by two nested rectangles. A pair of vertices on the boundary of the same rectangle are designated as terminals for each of the paths. The number of terminals lying on each boundary vertex is determined by the degree of the boundary vertex. Every vertex of degree 2 has either 0 or 2 terminals lying on ...

A graph is 1-planar if it can be embedded in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree 5 and girth 4 contains (1) a 5-vertex adjacent to an ≤ 6-vertex, (2) a 4-cycle whose every vertex has degree at most 9, (3) a K1,4 with all vertices having degree at most 11.